pros and cons of factoring quadratic equations

Pros and cons of factoring quadratic equations how to#

Importantly, these are all the solutions. Note that the solutions are the numbers we subtract from x, i.e., not − 3and − 4, but 3and 4. So, the xthat work are precisely those where x − 3 = 0(which is x = 3), or x − 4 = 0(which is x = 4). Then by dividing both sides of the equation a b = 0by a, we get b = 0, contradiction.) (The formal justification of this zero-product property uses the basic axiom that you can divide by any nonzero number: suppose for contradiction that a b = 0with both aand bnonzero. For example, suppose we want to find all xsuch that x 2 − 7 x + 12 = 0.We already know that this is the same (has exactly the same solutions) as ( x − 3 ) ( x − 4 ) = 0.The only way for two numbers to multiply to zero is if one (or both) are zero. The reason it is useful to know what happens when multiplying is because if we can do this in reverse, we can solve quadratic equations. Since we had both − 3and + 3, the + 3 uand − 3 uterms canceled out, giving us a difference of squares. Here's another example: ( u − 3 ) ( u + 3 ) = u 2 + 3 u − 3 u − 9 = u 2 − 9. The key takeaway is that the − 7in the − 7 xcomes from adding together − 3and − 4, and the 12comes from multiplying together − 3and − 4. First, we use the distributive rule to multiply (also called FOIL): ( x − 3 ) ( x − 4 ) = x 2 − 4 x − 3 x + 12 = x 2 − 7 x + 12. Let's start by reviewing the facts that are usually taught to introduce quadratic equations. The text discussion below goes a bit deeper and includes commentary which may be useful for teachers. This video is a self-contained practical lesson that walks through many examples with each logical step explained. It also shows a clean reduction of the problem from solving a standard quadratic, to a classical problem solved by the Babylonians and Greeks. It uses my sign convention and my own logical steps (as opposed to using Savage's version) in order to be logically sound, and also because I think my choice is helpful for understanding the deeper relationship between a quadratic and its solutions. The presentation below is based on the approach in my originally posted article, but goes further. Explanation of Quadratic Method, by Example

Since I still have not seen any previously-existing book or paper which states this type of method in a way that is suitable for first-time learners (avoiding advanced knowledge) and justifies all steps clearly and consistently, I chose to share it to provide a safely referenceable version. The related work page compares the method described by Savage, with the method that I proposed.

In particular, my approach's avoidance of an extra assumption in Completing the Square was not achieved by Savage's method. His approach overlapped in almost all calculations, with a pedagogical difference in choice of sign, but had a difference in logic, as (possibly due to a friendly writing style which leaves some logic up for interpretation) it appears to use the additional (true but significantly more advanced) fact that every quadratic can be factored into two linear factors, or has some reversed directions of implication that are not technically correct. The combination of these steps is something that anyone could have come up with, but after releasing this webpage to the wild, the only previous reference that surfaced, of a similar coherent method for solving quadratic equations, was a nice article by mathematics teacher John Savage, published in The Mathematics Teacher in 1989. The individual steps of this method had been separately discovered by ancient mathematicians. Known thousands of years ago (Babylonians, Greeks)

Thus − B 2 ± uwork as rand s, and are all the roots.

Two numbers sum to − Bwhen they are − B 2 ± u.

If you find rand swith sum − Band product C, then x 2 + B x + C = ( x − r ) ( x − s ), and they are all the roots.

Alternative Method of Solving Quadratic Equations

One night in September 2019, while brainstorming different ways to think about the quadratic formula, I was surprised to come up with a simple method of eliminating guess-and-check from factoring that I had never seen before.

Pros and cons of factoring quadratic equations how to#

I've recently been systematically thinking about how to explain school math concepts in more thoughtful and interesting ways, while creating my Daily Challenge lessons.